Average Error: 0.1 → 0.2
Time: 4.0s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[1 \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right) + \left(-m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
1 \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right) + \left(-m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)
double f(double m, double v) {
        double r9914 = m;
        double r9915 = 1.0;
        double r9916 = r9915 - r9914;
        double r9917 = r9914 * r9916;
        double r9918 = v;
        double r9919 = r9917 / r9918;
        double r9920 = r9919 - r9915;
        double r9921 = r9920 * r9916;
        return r9921;
}
↓
double f(double m, double v) {
        double r9922 = 1.0;
        double r9923 = m;
        double r9924 = r9922 - r9923;
        double r9925 = v;
        double r9926 = r9924 / r9925;
        double r9927 = -r9922;
        double r9928 = fma(r9923, r9926, r9927);
        double r9929 = r9922 * r9928;
        double r9930 = -r9923;
        double r9931 = r9930 * r9928;
        double r9932 = r9929 + r9931;
        return r9932;
}

Error

Bits error versus m

Bits error versus v

Derivation

  1. Initial program 0.1

    \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.1

    \[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{\color{blue}{1 \cdot v}} - 1\right) \cdot \left(1 - m\right)\]

  4. Applied times-frac0.2

    \[\leadsto \left(\color{blue}{\frac{m}{1} \cdot \frac{1 - m}{v}} - 1\right) \cdot \left(1 - m\right)\]

  5. Applied fma-neg0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{m}{1}, \frac{1 - m}{v}, -1\right)} \cdot \left(1 - m\right)\]
  6. Using strategy rm

  7. Applied sub-neg0.2

    \[\leadsto \mathsf{fma}\left(\frac{m}{1}, \frac{1 - m}{v}, -1\right) \cdot \color{blue}{\left(1 + \left(-m\right)\right)}\]

  8. Applied distribute-lft-in0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{m}{1}, \frac{1 - m}{v}, -1\right) \cdot 1 + \mathsf{fma}\left(\frac{m}{1}, \frac{1 - m}{v}, -1\right) \cdot \left(-m\right)}\]

  9. Simplified0.2

    \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} + \mathsf{fma}\left(\frac{m}{1}, \frac{1 - m}{v}, -1\right) \cdot \left(-m\right)\]

  10. Simplified0.2

    \[\leadsto 1 \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right) + \color{blue}{\left(-m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)}\]

  11. Final simplification0.2

    \[\leadsto 1 \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right) + \left(-m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))